I live in England, and here cheese rarely has animal rennet, with a few exceptions like parmasan. I'm not sure how you would find vegetarian cheese in America though, since as far as I an tell they don't put whether something is vegetarian on the label of most things.

All of these cheeses melt as well! Daiya is my favorite melted, Tofutti is my favorite on sandwiches.

Daiya rocks. Wayfare is also great, its a grain based cheese, but its a spreadable cheese that comes in many flavors called We can't say its cheese. They also make puddings, sour cream, and I think a cream cheese.

Just read the ingredients. I don't know where you're from, but where I live I barely ever have trouble finding cheese without animal rennet in it. I buy cheeses which list "vegetarian rennet" - no it may not be vegan, but if you're eating dairy then you're not a vegan anyway...?

Cottage cheese may or may not have rennet in it - you'd have to check the ingredients. Yogurt doesn't contain rennet as far as I know. However, you have to watch out for gelatine in both products. If possible I only buy brands which say explicitly that they're suitable for vegetarians.